Numerical Methods · Unit 6
Numerical Integration
Exam-focused notes for Numerical Integration (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 8 solved past questions from this unit.
What this unit covers
- Trapezoidal rule and composite trapezoidal rule
- Simpson's 1/3 rule and composite Simpson's 1/3 rule
- Simpson's 3/8 rule and composite Simpson's 3/8 rule
- Double integration using Simpson's rules
- Applications of numerical integration
- Algorithm and program for numerical integration
Simpson's 1/3 rule and composite Simpson's 1/3 rule
Integrate $\int_{0}^{3} (2x^3 + 1) dx$ using Simpson's $\frac{1}{3}$ rule with $n=6$. [5]
- $f(x) = 2x^3 + 1$ - Limits: $a = 0$, $b = 3$ - Number of intervals: $n = 6$ $$h = \frac{b-a}{n} = \frac{3-0}{6} = 0.5$$ $i$ $xi$ $f(xi) = 2xi^3 + 1$ Coeff Product -------------------------------- 0 0.0 $2(0)+1 = 1.000$ 1 1.000 1 0.5 $2(0.125)+1 = 1.250$ 4...
Full solved answer →Double integration using Simpson's rules
Solve the double integration using Simpson's 1/3 rule. $$\int_{2}^{2.6} \int_{4}^{4.4} \frac{dxdy}{xy}$$ [5]
$$I = \int{2}^{2.6} \int{4}^{4.4} \frac{dx\,dy}{xy}$$ - Inner variable $x$: limits $4$ to $4.4$ - Outer variable $y$: limits $2$ to $2.6$ - Integrand: $f(x,y) = \dfrac{1}{xy}$ Using $n = 2$ subintervals in each direction: - $h = \dfrac{4.4 - 4}{2} = 0.2$, g...
Full solved answer →Trapezoidal rule and composite trapezoidal rule
Why Numerical Integration is required? Compute the integral: $I=\int_{-1}^{1} e^x dx$ using composite trapezoidal rule for n = 4. [5]
Numerical integration (numerical quadrature) is required because: 1. No closed-form antiderivative exists for many functions such as $e^{-x^2}$ or $\frac{\sin x}{x}$. 2. The function is known only at discrete points (tabulated/experimental data), with no ex...
Full solved answer →Solve the following integral using trapezoidal rule form = 8, $l=\int_{2}^{4} (x^4 + 1)dx$. [5]
- Integral: $l = \int2^4 (x^4 + 1)\,dx$ - $a = 2$, $b = 4$ - Number of subintervals: $n = 8$ Step size: $$h = \frac{b-a}{n} = \frac{4-2}{8} = 0.25$$ Table of values with $f(x) = x^4 + 1$: $i$ $xi$ $xi^4$ $f(xi)$ ------------------------------- 0 2.00 16.000...
Full solved answer →Find following integral using composite trapezoidal rule using 2 segments (k=2) and 4 segments (k=4). $$\int_{2}^{8} (x+3)^2 dx$$ [5]
Integral: $\displaystyle\int2^8 (x+3)^2\,dx$ - Lower limit: $a = 2$ - Upper limit: $b = 8$ - Integrand: $f(x) = (x+3)^2$ - Cases: $k = 2$ segments and $k = 4$ segments All data present. $$\inta^b f(x)\,dx \approx \frac{h}{2}\left[f(x0) + 2\sum{i=1}^{k-1}f(x...
Full solved answer →Algorithm and program for numerical integration
What do you mean by numerical integration? Write any one application of numerical integration. Write an algorithm and c program to implement multi-segment trapezoidal rule.[10]
--- Numerical Integration (also called numerical quadrature) is the process of computing the approximate value of a definite integral using numerical methods when: - The integrand f(x) is too complex to integrate analytically, or - The function is given onl...
Full solved answer →Simpson's 3/8 rule and composite Simpson's 3/8 rule
Simpson's 3/8 Rule Integration Problem
The simple Simpson's 3/8 rule fits a single cubic polynomial over 3 sub-intervals (4 points). Its limitations: - It applies only to exactly 3 sub-intervals. A dataset with many points cannot be handled directly. - Fitting one cubic over a wide interval prod...
Full solved answer →Derive the formula for integration using simpsons 3/8 rule. Use Secant Method to estimate the root of equation with initial estimate x₁ = 4 and x₂ = 2, $x^2 - 4x - 10 = 0$. [10]
Divide $[a,b]$ into 3 equal sub-intervals with 4 points: $$x0=a,\quad x1=a+h,\quad x2=a+2h,\quad x3=a+3h=b,\qquad h=\frac{b-a}{3}$$ Let $yi=f(xi)$. Use Newton's forward interpolation with $x=x0+th$, $dx=h\,dt$: $$f(x)\approx y0+t\Delta y0+\frac{t(t-1)}{2!}\...
Full solved answer →Make Unit 6 stick
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